1. Field of the Invention
The present invention is directed to an image reconstruction method for a multi-row detector computed tomography apparatus operable in a helical/spiral scan mode.
2. Description of the Prior Art
For image reconstruction from data of a simple complete circular revolution of a multi-row detector (MZD) CT scanner, the known Feldkamp algorithm is suitable (see: L. A. Feldkamp, L. C. Davis, J. W. Kress, "Practical cone-beam algorithm," J. Opt. Soc. Am. A, vol. 1, no. 6, June 1984). A generalization of this algorithm by Wang et al. (see: G. Wang, T. H. Lin, P. C. Cheng, D. M. Shinozaki, "A General Cone-Beam Reconstruction Algorithm," IEEE Transactions on Medical Imaging, vol. 12, no. 3, September 1993), can reconstruct images from spiral scans, however, this algorithm has some serious limitations.
The reconstruction algorithm specified by Wang et al. for a planar detector is formulated herein for the case of a cylindrical detector. FIG. 1 illustrates the construction of the MZD-CT scanner. The multi-row detector 1 is constructed as a cylinder surface. It has several parallel detector rows, each of which is formed from a series of detector elements. The radius of this cylinder is R.sub.f +R.sub.d, i.e. the focus 2 is located on the cylinder axis. The focus 2 describes a spiral path with the focus path radius R.sub.f. R.sub.d is the distance of the detector 1 to the axis of rotation z. The projection angle (angle of rotation of the gantry with the X-ray source and the detector) is designated .alpha.. .beta..sub.m and .zeta..sub.Det,q identify a particular detector element, and thereby a particular beam from the bundle of rays. .beta..sub.m is the fan angle of the beam, and q is the index of the relevant detector row. The z coordinates of the focus are designated z.sub.F, and the spiral has the helical slope slope, described by: EQU z.sub.F =z.sub.F,O +n.DELTA..alpha..multidot.slope (1)
wherein .DELTA..alpha. is the projection angle increment between successive projections, n is the number of the relevant projection, and z.sub.F,O is the z start position of the spiral.
.zeta..sub.Det =z-z.sub.F specifies the axial position of a point on the beam in relation to the focus, of which point the projected distance from the focus in the x-y plane is already R.sub.f +R.sub.d. This is at the same time the z position of the relevant detector element relative to the z position of the focus. The .zeta..sub.Det position of the detector row with index q is given by ##EQU1## wherein .DELTA..zeta..sub.Det is the distance between two detector rows in the z direction N.sub.rows is the number of detector rows, and AQ is the alignment in the .zeta. direction. The distance, projected into the rotational center, between two detector rows in the z direction ##EQU2## or the .zeta. coordinates projected thereto ##EQU3## is also often used. The logarithmized attenuation values, i.e. the line integrals measured by the detectors via the attenuation coefficients of the object, are designated p(.alpha..sub.n,.beta..sub.m,.zeta..sub.Det,q), or, in abbreviated form, p(n,m,q). Thereby ##EQU4## as well as ##EQU5## wherein N.sub.p,2.pi. is the number of projections per 2.pi. revolution, and N.sub.P is the total number of projections present. N is the number of channels in one row of the detector. For simplicity, it is assumed herein that N is even. AM is the alignment in the fan angle.
The reconstruction method according to Wang et al. in the formulation for cylindrical detectors contains the following steps:
1. Weighting and convolution of the projections ##EQU6## wherein cos .beta..sub.m is the cos weighting of the data in the row direction required in the fan reconstruction. The convolution takes place only along the rows of the projection, i.e. the operation is carried out independently for all q. The convolution kernel g.sub.m is e.g. the known cotangent kernel ##EQU7## and the distribution g(.beta.) is defined by ##EQU8## 2. Weighted back-projections of the convoluted projections
The back-projection following the convolution is described by ##EQU9## wherein n.sub.0 is a summation index that carries out the summation over all projection angles of a complete revolution. As a rule, in a spiral data set there are several different projections at the same projection angle .alpha..sub.n.sbsb.0, namely one in each "winding" of the spiral. By this means, for the unambiguous identification of a projection, besides the projection angle determined by n.sub.0, the number .lambda. of the "winding" containing it is indicated. In general, in the back-projection a voxel of the reconstruction volume lies in several projections with the same n.sub.0 (thus the same projection angle), but different .lambda.. Thus, an index .lambda. must be selected that determines the projection to be used for the back-projection of this voxel. In the Wang algorithm, this .lambda. is selected so that the focus position of the projection used has the smallest possible distance in the axial direction from the relevant voxel.
The expression ##EQU10## is what is called the 1/r.sup.2 weighting that is required in the back-projection of the convolved measurement values in the fan reconstruction.
The indices m and q result from the projection of a voxel V(x,y,z), to be reconstructed, of the reconstruction volume from the focus to the detector. Let the voxel projected to the detector have the coordinates B and .xi..sub.Det. The following is then obtained for the indices m and q: ##EQU11## or, ##EQU12## m is thereby independent of z. Since the projection of a voxel from the focus into the detector generally does not strike precisely on a detector element, non-integer-number indices also arise in the evaluation of equations (9) and (10). For this reason, in the back-projection an interpolation takes place between the adjacent elements (e.g. four) of the convoluted projection (e.g.: bilinear interpolation).
In the Wang algorithm, it must be ensured that each voxel of the reconstruction volume receives a back-projection contribution from all directions x=0 . . . 2.PI. of a full revolution. As mentioned above, for each index n.sub.0 a revolution of the spiral is individually chosen for each voxel, from which turn the projection with the corresponding projection angle .alpha..sub.n.sbsb.0 is employed. The .lambda.(x,y,z,n.sub.0) are selected so that the selected projection is always the one lying nearest the relevant voxel in the z direction. The situation is illustrated in FIG. 2. In the back-projection under the projection angle .alpha..sub.n.sbsb.0 =0, for the voxel V.sub.1, the value of .lambda. is chosen equal to 0, while in contrast .lambda. is chosen equal to 1 for the voxel V.sub.2. Thus, for the back-projection into the voxel V.sub.1, the projection shown at left is used, and for the back-projection into the voxel V.sub.2, the projection shown at right is used, although for both voxels both projections would be possible.
The requirement that each voxel must receive back-projection contributions from all projection angles is the main disadvantage of the Wang algorithm. This requirement limits the pitch of the spiral, defined as the z-advance .DELTA.z.sub.2.pi. of the spiral per 2.pi. complete revolution of the gantry, normalized to the row distance of the detector rows .DELTA..zeta. projected into the rotational center. ##EQU13##
FIG. 2 illustrates this pitch limitation. A voxel V at the edge of the image field, with diameter D.sub.B, must receive contributions from all directions x=0 . . . 2.PI.. For this, the pitch cannot become larger than is shown in FIG. 2. The pitch is thus at a maximum when the beams (1) and (2) intersect exactly at the edge of the image field. The maximum pitch thereby results at: ##EQU14## For a particular gantry geometry, there results e.g. (R.sub.f =570 mm, R.sub.d =435 mm, D.sub.B =500 mm):
______________________________________ rows 3 4 5 6 7 ______________________________________ pitch.sub.max 1.1228 1.6842 2.2456 2.8070 3.3684 ______________________________________
If the pitch of the spiral is increased above this critical value, gaps arise in the sampling scheme (see FIG. 3, shading). For these shaded gaps, there are no projections under certain projection angles x that could supply a contribution in these areas in the back-projection. The original Wang algorithm thus cannot be used for the reconstruction of spirals with such large pitch values. A pitch of 2.25 for a 5-row detector is however too small for a practically usable application.
A second disadvantage of the Wang algorithm is that the back-projection is carried out projection-by-projection. For each projection angle, the supplement of a projection to a voxel, which projection is recorded under this projection angle, is determined by interpolation between the elements of the convolved projection that are adjacent to the projection of the relevant voxel from the focus into the detector. Accordingly, the range of the interpolation in the z direction in the rotational center is ##EQU15##